Theorem 2initoinv 18055

Description: Morphisms between two initial objects are inverses. (Contributed by AV, 14-Apr-2020.)

Hypotheses

Ref	Expression
initoeu1.c	⊢ (𝜑 → 𝐶 ∈ Cat)
initoeu1.a	⊢ (𝜑 → 𝐴 ∈ (InitO‘𝐶))
initoeu1.b	⊢ (𝜑 → 𝐵 ∈ (InitO‘𝐶))

Assertion

Ref	Expression
2initoinv	⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝐹(𝐴(Inv‘𝐶)𝐵)𝐺)

Proof of Theorem 2initoinv

Step	Hyp	Ref	Expression
1		eqid 2737	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		eqid 2737	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3		eqid 2737	. . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶)
4		initoeu1.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat)
5	4	3ad2ant1 1134	. . . . 5 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝐶 ∈ Cat)
6		initoeu1.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (InitO‘𝐶))
7		initoo 18052	. . . . . . 7 ⊢ (𝐶 ∈ Cat → (𝐴 ∈ (InitO‘𝐶) → 𝐴 ∈ (Base‘𝐶)))
8	4, 6, 7	sylc 65	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝐶))
9	8	3ad2ant1 1134	. . . . 5 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝐴 ∈ (Base‘𝐶))
10		initoeu1.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ (InitO‘𝐶))
11		initoo 18052	. . . . . . 7 ⊢ (𝐶 ∈ Cat → (𝐵 ∈ (InitO‘𝐶) → 𝐵 ∈ (Base‘𝐶)))
12	4, 10, 11	sylc 65	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ (Base‘𝐶))
13	12	3ad2ant1 1134	. . . . 5 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝐵 ∈ (Base‘𝐶))
14		simp3 1139	. . . . 5 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵))
15		simp2 1138	. . . . 5 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴))
16	1, 2, 3, 5, 9, 13, 9, 14, 15	catcocl 17728	. . . 4 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → (𝐺(⟨𝐴, 𝐵⟩(comp‘𝐶)𝐴)𝐹) ∈ (𝐴(Hom ‘𝐶)𝐴))
17	1, 2, 4	initoid 18046	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ∈ (InitO‘𝐶)) → (𝐴(Hom ‘𝐶)𝐴) = {((Id‘𝐶)‘𝐴)})
18	6, 17	mpdan 687	. . . . . . 7 ⊢ (𝜑 → (𝐴(Hom ‘𝐶)𝐴) = {((Id‘𝐶)‘𝐴)})
19	18	3ad2ant1 1134	. . . . . 6 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → (𝐴(Hom ‘𝐶)𝐴) = {((Id‘𝐶)‘𝐴)})
20	19	eleq2d 2827	. . . . 5 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → ((𝐺(⟨𝐴, 𝐵⟩(comp‘𝐶)𝐴)𝐹) ∈ (𝐴(Hom ‘𝐶)𝐴) ↔ (𝐺(⟨𝐴, 𝐵⟩(comp‘𝐶)𝐴)𝐹) ∈ {((Id‘𝐶)‘𝐴)}))
21		elsni 4643	. . . . 5 ⊢ ((𝐺(⟨𝐴, 𝐵⟩(comp‘𝐶)𝐴)𝐹) ∈ {((Id‘𝐶)‘𝐴)} → (𝐺(⟨𝐴, 𝐵⟩(comp‘𝐶)𝐴)𝐹) = ((Id‘𝐶)‘𝐴))
22	20, 21	biimtrdi 253	. . . 4 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → ((𝐺(⟨𝐴, 𝐵⟩(comp‘𝐶)𝐴)𝐹) ∈ (𝐴(Hom ‘𝐶)𝐴) → (𝐺(⟨𝐴, 𝐵⟩(comp‘𝐶)𝐴)𝐹) = ((Id‘𝐶)‘𝐴)))
23	16, 22	mpd 15	. . 3 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → (𝐺(⟨𝐴, 𝐵⟩(comp‘𝐶)𝐴)𝐹) = ((Id‘𝐶)‘𝐴))
24		eqid 2737	. . . 4 ⊢ (Id‘𝐶) = (Id‘𝐶)
25		eqid 2737	. . . 4 ⊢ (Sect‘𝐶) = (Sect‘𝐶)
26	1, 2, 3, 24, 25, 5, 9, 13, 14, 15	issect2 17798	. . 3 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → (𝐹(𝐴(Sect‘𝐶)𝐵)𝐺 ↔ (𝐺(⟨𝐴, 𝐵⟩(comp‘𝐶)𝐴)𝐹) = ((Id‘𝐶)‘𝐴)))
27	23, 26	mpbird 257	. 2 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝐹(𝐴(Sect‘𝐶)𝐵)𝐺)
28	1, 2, 3, 5, 13, 9, 13, 15, 14	catcocl 17728	. . . 4 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → (𝐹(⟨𝐵, 𝐴⟩(comp‘𝐶)𝐵)𝐺) ∈ (𝐵(Hom ‘𝐶)𝐵))
29	1, 2, 4	initoid 18046	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ∈ (InitO‘𝐶)) → (𝐵(Hom ‘𝐶)𝐵) = {((Id‘𝐶)‘𝐵)})
30	10, 29	mpdan 687	. . . . . . 7 ⊢ (𝜑 → (𝐵(Hom ‘𝐶)𝐵) = {((Id‘𝐶)‘𝐵)})
31	30	3ad2ant1 1134	. . . . . 6 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → (𝐵(Hom ‘𝐶)𝐵) = {((Id‘𝐶)‘𝐵)})
32	31	eleq2d 2827	. . . . 5 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → ((𝐹(⟨𝐵, 𝐴⟩(comp‘𝐶)𝐵)𝐺) ∈ (𝐵(Hom ‘𝐶)𝐵) ↔ (𝐹(⟨𝐵, 𝐴⟩(comp‘𝐶)𝐵)𝐺) ∈ {((Id‘𝐶)‘𝐵)}))
33		elsni 4643	. . . . 5 ⊢ ((𝐹(⟨𝐵, 𝐴⟩(comp‘𝐶)𝐵)𝐺) ∈ {((Id‘𝐶)‘𝐵)} → (𝐹(⟨𝐵, 𝐴⟩(comp‘𝐶)𝐵)𝐺) = ((Id‘𝐶)‘𝐵))
34	32, 33	biimtrdi 253	. . . 4 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → ((𝐹(⟨𝐵, 𝐴⟩(comp‘𝐶)𝐵)𝐺) ∈ (𝐵(Hom ‘𝐶)𝐵) → (𝐹(⟨𝐵, 𝐴⟩(comp‘𝐶)𝐵)𝐺) = ((Id‘𝐶)‘𝐵)))
35	28, 34	mpd 15	. . 3 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → (𝐹(⟨𝐵, 𝐴⟩(comp‘𝐶)𝐵)𝐺) = ((Id‘𝐶)‘𝐵))
36	1, 2, 3, 24, 25, 5, 13, 9, 15, 14	issect2 17798	. . 3 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → (𝐺(𝐵(Sect‘𝐶)𝐴)𝐹 ↔ (𝐹(⟨𝐵, 𝐴⟩(comp‘𝐶)𝐵)𝐺) = ((Id‘𝐶)‘𝐵)))
37	35, 36	mpbird 257	. 2 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝐺(𝐵(Sect‘𝐶)𝐴)𝐹)
38		eqid 2737	. . . 4 ⊢ (Inv‘𝐶) = (Inv‘𝐶)
39	1, 38, 4, 8, 12, 25	isinv 17804	. . 3 ⊢ (𝜑 → (𝐹(𝐴(Inv‘𝐶)𝐵)𝐺 ↔ (𝐹(𝐴(Sect‘𝐶)𝐵)𝐺 ∧ 𝐺(𝐵(Sect‘𝐶)𝐴)𝐹)))
40	39	3ad2ant1 1134	. 2 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → (𝐹(𝐴(Inv‘𝐶)𝐵)𝐺 ↔ (𝐹(𝐴(Sect‘𝐶)𝐵)𝐺 ∧ 𝐺(𝐵(Sect‘𝐶)𝐴)𝐹)))
41	27, 37, 40	mpbir2and 713	1 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝐹(𝐴(Inv‘𝐶)𝐵)𝐺)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  {csn 4626  ⟨cop 4632   class class class wbr 5143  ‘cfv 6561  (class class class)co 7431  Basecbs 17247  Hom chom 17308  compcco 17309  Catccat 17707  Idccid 17708  Sectcsect 17788  Invcinv 17789  InitOcinito 18026

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-cat 17711  df-cid 17712  df-sect 17791  df-inv 17792  df-inito 18029

This theorem is referenced by:  initoeu1  18056